Talk by Mostafa Sabri

The discrete Laplacian of an infinite, Z^d-periodic graph (e.g. Z^d itself, the honeycomb lattice, infinite strips, face-centered cubic lattices etc.) has a band spectrum which is typically absolutely continuous, but for certain geometries, infinitely degenerate eigenvalues appear, which are known as flat bands. They have corresponding eigenvectors of compact support. Flat bands have been extensively studied by physicists in recent years, with hundreds of papers on the subject. Mathematically they have been much less explored, and many fundamental questions remain completely open. I will introduce this exciting topic, give some recent results with Pierre Youssef, and discuss some open problems.